24.3 Generalizations to Higher Dimensions

Everything discussed above applies equally well to integrals in any number of Euclidean dimension.

Thus you will encounter
volume integrals in three dimensions,
for these the volume element
dV
can be expressed in terms of variables
s,t
and
u
according to the expression
dV=dxdydz=Jdsdtdu
where
J
,
the Jacobian, is the absolute value of the determinant of the partial derivatives of the variables
x,y and
z with respect to
s,t andu
.

There are also integrals over "phase space" in which there are 3 position and 3 momentum variables for each particle, integrals over space and time, and integrals over the positions of each of many particles.

In higher dimensions, the analog of volume is called
hyper-volume,
and the analog of a surface is called a
hyper-surface.
It is a region in one dimension less than that of the whole space.

A flat hyper-surface (called a
hyper-plane
) has a normal direction, which is
perpendicular to all differences between points within it.

Again we can define integrals over hyper-surfaces of the normal component of a vector
w&LongRightArrow;
, by defining them locally over infinitesimal flat surfaces as done above.

Again we find that the integrals so defined become integrals of determinants of
w&LongRightArrow;
and the partials of the coordinates on the surface with respect to the parameters by which the surface is defined, exactly as happened above.

Again
changes of coordinate systems cause changes in hypervolume elements
that can be described by Jacobians exactly as before. The only difference is
that the relevant determinants are have higher dimension.

Exercise 24.5

Find the Jacobian going from cylindric coordinates to spherical coordinates,
that is find relation between
drdθdz anddθdρdφ
.